Robust covariance estimates based on resampling

Many asymptotic covariance estimates are generated using uncontaminated model distributions and thus are often based in part on the information matrix. Such covariance estimators have a low breakdown point (Donoho and Huber, 1983; Huber, 1981; Hampel et al., 1986, p. 98; Lopuhaa and Rousseeuw, 1991), even if the estimate itself has a high breakdown point. These covariance estimates will not be reliable when there are outliers present. As alternative estimates of variability for robust estimators, we consider using the bootstrapped or jackknifed sample covariance matrix. As will be shown in this paper, the bootstrapped sample covariance matrix can have a breakdown point of l/n regardless of the breakdown point of the estimator; thus both bootstrapped and asymptotic covariance estimates may be heavily influenced by outliers even if the original estimate is not. This is not the case for the jackknifed sample covariance matrix. If an estimate is not heavily influenced by outliers, its jackknifed sample covariance estimate is not likely to be heavily influenced by outliers. On the other hand, the jackknife, but not the bootstrap, may have a low breakdown point because the covariance estimate can often be shifted to zero by shifting a few points.